LABORATOIRE DE PHYSIQUE THEORIQUE DE LA MATIERE CONDENSEE

Counting the equilibria of a directed polymer in a random medium and Anderson localisation

I will discuss a new connection between two different problems: the counting of equilibria of a directed polymer in a random medium (DPRM) and the problem of Anderson localisation for the 1D Schrödinger equation. Using the Kac-Rice formula, it is possible to express the mean number of equilibria of a DPRM in terms of functional determinants. In the one dimensional situation, these functional determinants can be calculated thanks to the Gelfand-Yaglom method, showing that the mean number of equilibria of the DPRM growth exponentially with the length of the polymer, with a rate controlled by the generalized Lyapunov exponent (GLE) of the localisation problem (cumulant generating function of the log of the wave function). The GLE is solution of a spectral problem studied by combining numerical approaches and WKB-like approximation. Furthermore, the formalism can be extended in order to obtain the number of equilibria at fixed energy, providing the (annealed) distribution of the energy density of the line over the equilibria.

In this talk, I will present newly developed physics-inspired methods for the solution of counting constraint satisfaction problems (#CSPs). #CSP instances can be reformulated as interacting models whose zero-temperature partition function represents the volume of the solution manifold. I will introduce practical methods to compute such partition functions based on tensor network contraction. In this formulation, computational complexity can be viewed as a manifestation of quantum entanglement, and controlling the growth of entanglement throughout tensor network contraction can yield a significant computation speedup. Using some hard counting problems as benchmarks, I will demonstrate that tensor network methods can be a useful tool for solving some hard classes of #CSPs. I will conclude with an outline of ongoing work on extensions of this framework, such as the simulation of existing and near-term quantum circuits.

Dirac systems and topological materials are two rapidly growing and evolving fields in modern condensed matter physics, with a very long history from soliton and quantum Hall physics in the early eighties; and also a more recent history dating from the isolation of graphene in 2004 and the prediction of topological insulators in 2005. In these lectures, i will discuss the topogical aspects of non-interacting fermions on lattices and their relation to Dirac fermions. The goal will be to introduce the basic concepts (topological invariants, quantized electromagnetic response, bulk-boundary correspondance, Dirac fermions, symmetries,…) on simple, yet very rich, models with a progression from one-dimensional (1D) chains to three-dimensional (3D) crystals.
In the first lecture (Thursday 4/10/18), i will use the Su-Schrieffer-Heeger and Rice-Mele models to introduce the concepts of Berry-Zak phase, winding numbers and zero energy end states in 1D. Then we will discuss how those ideas can be transposed and extended to 2D lattices, using the Bernevig-Hughes-Zhang model as a typical example of a Chern insulator. The relations between Berry curvature, Chern number, quantized Hall effect will be detailled.
The second lecture (Friday 5/10) will treat further aspects of 2D topological insulators with an emphasis on graphene (Haldane and Kane-Mele models) and a discussion of topological invariants in presence of time-reversal symmetry. I will conclude by a short list of experimental realisations of 1D and 2D topological systems. If time allows, i will discuss briefly 3D topological insulators and semimetals (Dirac and Weyl semimetals).

Consider a statistical system evolving on a state space of graphical structures, such as e.g. a social network system. Given a set of transitions on such a system, where each transition consists of a local transformation pattern applied at random to the system's state (e.g. adding a new edge, deleting an edge,...), one may define a continuous-time Markov chain in order to study the stochastic evolution of the system. Our novel approach to this problem involves an extension of Doi's description of chemical reaction systems in terms of boson creation and annihilation operators (which later evolved into the Doi-Peliti formalism) to a general stochastic mechanics framework based on the idea of so-called rule algebras. Assuming no prior familiarity with the underlying concept of graph rewriting and related mathematics, I will give an introduction to the formalism and present a number of application examples.

Lecture I: The Beresinskii-Kosterlitz-Thouless transition: the two-dimensional world and its peculiarities
More than 40 years after the seminal work by Berezinskii, Kosterlitz and Thouless the
BKT transition remains one of the most spectacular phenomena in condensed matter
systems, as it has been acknowledged by the 2016 Nobel Prize. Even though it was originally formulated within the context of the two dimensional XY model for classical spins, it represents the paradigm for the superfluid transition in two dimensions. As such, it has been the subject of an intense theoretical and experimental investigation in a variety of systems, ranging from thin films of superconductors to artificial heterostructures and cold atoms.
In the first lecture I will give an introduction to the basic mathematical ingredients needed to understand the occurrence of a BKT transition within the context of the classical 2D XY model. After discussing the difference between order and rigidity for a second-order phase transition, I will discuss the peculiar role of vortices in 2D and I will derive the mapping onto the Coulomb-gas model. Finally, I will sketch the main outcomes of the renormalization-group approach for the BKT phase transition.

Lecture II: Applications to superfluids. What we should expect to see in real systems?
The Beresinskii-Kosterlitz-Thouless transition is expected to describe the metal-to-superconductor thermal transition in quasi-2D systems. However, despite many efforts along the years its signatures remain rather elusive. In this second lectures I will give an overview of the numerous attempts we made along the years to identify the mechanisms which may hinder a clear-cut observation of BKT physics in 2D films of conventional and unconventional superconductors. In particular I will discuss the role of the vortex-core energy and of the spontaneous inhomogeneity of the superconducting background which naturally emerge in disordered thin films. These effects must be seriously taken into account while addressing the famous universal jump of the superfluid density, the non-linear IV characteristics near Tc, or the paraconductivity effects while approaching it from above. Finally, I will make a comparison with some recent results within the context of cold atoms, underlying differences and analogies between the two classes of systems.